69 problems

Terao's conjecture. Freeness is determined solely by the intersection lattice $\mathcal{L}$; equivalently, freeness is constant on $\mathcal{R}(\mathcal{L})$.

Let $C_{1}, C_{2}$ be reduced curves in $\mathbb{P}^{2}_{\mathbb{C}}$ whose irreducible components are smooth and whose singularities are ordinary quasi-homogeneous singularities.…

Maximal Tjurina existence conjecture. For every integer $d\geq 3$ and every integer $r$ with $d/2\leq r\leq d-1$, there are maximal Tjurina curves of type $(d,r)$. Moreover, for…

Let a conic-line arrangement be an arrangement $\mathcal{CL}$ of lines and smooth conics in the complex projective plane with ordinary singularities, and let $\gamma(\mathcal{CL})$…

Let $C$ be a reduced plane curve in $cmathbb{P}^2$. A point $p\in C$ is modular if the central projection from $p$ induces a locally trivial fibration of the complement…

Let $\mathfrak{\mathcal{C}}_{0}$ be the arrangement of six lines whose normals are the columns of the matrix $M_{t}$ for $t=\pm2\pm\sqrt{5}\in\mathbb{P}^{1}\setminus U$, and let…

Let $C$ be a line arrangement that is not a union of concurrent lines, and let ${\alpha}$ be a root of its Alexander polynomial $\Delta^1_C(t)$. Suppose that the order of…

Let an arrangement of $n$ lines in the projective plane be non-pencil, meaning that the lines are not all concurrent. A simple point is an intersection point incident with exactly…

Four-line generation conjecture.

Supersolvable arrangement double-point conjecture. One has

Let $m$ lines in the plane intersect in a single point, and let $b4_n$ denote the associated invariant for the arrangement, with $n\geq 0$. Conjecture. For every $n\geq 0$, … This…

A multinet in $\mathbb{CP}^2$ is a line arrangement with the multinet structure defined in the paper, and a net is the corresponding structure with all line multiplicities equal to…

Let $F(x,y)$ be a polynomial whose zero set is a plane curve consisting of the product of $d$ real lines. Consider the critical points of $F$ having the same non-zero critical valu…

Cohen–Suciu conjecture. The incidence lattice determines both the affine and the projective fundamental groups of a real line arrangement.

Let $S=\mathbb C[x,y,z]$, let $\mathcal A:f=0$ be a line arrangement in $\mathbb P^2$, and write $d=\deg f$. Its minimal degree of a Jacobian relation is … where…

Finiteness conjecture. The number of infinite series of type described by Forge and Bartholdi arising from stretchable optimal arrangements is finite.

Line-cover conjecture. One has

Let a line arrangement be a finite collection of lines in the complex projective plane, and let $\gamma(\mathcal{L})$ denote its characteristic number. For a real number…

Numerical Terao's conjecture. If $\mathcal{L}_{1},\mathcal{L}_{2}\subset\mathbb{P}^{2}_{\mathbb{F}}$ are line arrangements satisfying

For a reduced plane curve $C=\{f=0\}$ in $\mathbb{P}^{2}_{\mathbb{C}}$, let $\nu(C)$ denote its defect. A line arrangement is a union of finitely many lines in…

Let $\mathcal A:f=0$ be a line arrangement of degree $d$ in the complex projective plane. Let $\operatorname{mdr}(f)$ denote the minimal degree of a Jacobian relation for $f$, and…

21-diameter conjecture. The maximal number of real diameters obtainable by a small deformation of three real lines is $21$.

Diameter upper-bound conjecture. The number of diameters of $\mathcal R$ does not exceed

Curvature-extrema conjecture. The number of minima of the curvature on $\mathcal R$ plus the number of inflection points on $\mathcal R$ equals the number of bounded edges of…

Let $\mathcal{L}$ be a sporadic simplicial line arrangement in $\mathbb{P}^{2}_{\mathbb{R}}$, meaning a simplicial line arrangement outside the known infinite families, and let $d$…